Question: Solve for $x$ : $ 5|x + 1| - 6 = 4|x + 1| + 9 $
Subtract $ {4|x + 1|} $ from both sides: $ \begin{eqnarray} 5|x + 1| - 6 &=& 4|x + 1| + 9 \\ \\ { - 4|x + 1|} && { - 4|x + 1|} \\ \\ 1|x + 1| - 6 &=& 9 \end{eqnarray} $ Add ${6}$ to both sides: $ \begin{eqnarray} 1|x + 1| - 6 &=& 9 \\ \\ { + 6} &=& { + 6} \\ \\ 1|x + 1| &=& 15 \end{eqnarray} $ Simplify: $ |x + 1| = 15$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 1 = -15 $ or $ x + 1 = 15 $ Solve for the solution where $x + 1$ is negative: $ x + 1 = -15 $ Subtract ${1}$ from both sides: $ \begin{eqnarray} x + 1 &=& -15 \\ \\ {- 1} && {- 1} \\ \\ x &=& -15 - 1 \end{eqnarray} $ $ x = -16 $ Then calculate the solution where $x + 1$ is positive: $ x + 1 = 15 $ Subtract ${1}$ from both sides: $ \begin{eqnarray} x + 1 &=& 15 \\ \\ {- 1} && {- 1} \\ \\ x &=& 15 - 1 \end{eqnarray} $ $ x = 14 $ Thus, the correct answer is $x = -16 $ or $x = 14 $.